A triangle is one of the most fundamental shapes in geometry, a closed two-dimensional figure with three straight sides and three angles. Its simplicity belies a rich set of properties and classifications that are crucial in mathematics, engineering, physics, and many other fields.

Here are the key properties of a triangle:

Basic Properties

1. Three Sides, Three Vertices, Three Angles: Every triangle, by definition, has exactly three straight line segments (sides), three points where these segments meet (vertices), and three angles formed at these vertices.

2. Angle Sum Property: The sum of the three interior angles of any triangle is always equal to 180 degree (or π radians). This is a foundational property.

   • For a triangle with angles ∠A, ∠B, and ∠C: ∠A+∠B+∠C=180 degree.

Classification by Side Lengths:

1. Equilateral Triangle:

   • Sides: All three sides are equal in length.

   • Angles: All three interior angles are equal, and each measures 60 degree.

   • It is also an equiangular triangle.

   • It has three lines of symmetry.

2. Isosceles Triangle:

   • Sides: At least two sides are equal in length.

   • Angles: The angles opposite the two equal sides are also equal. These are often called the "base angles."

   • It has at least one line of symmetry.

3. Scalene Triangle:

   • Sides: All three sides are of different lengths.

   • Angles: All three interior angles are of different measures.

   • It has no lines of symmetry.

Classification by Interior Angles:

1. Acute-angled Triangle:

   • Angles: All three interior angles are acute (i.e., less than 90 degree).

2. Right-angled Triangle:

   • Angles: Exactly one interior angle is a right angle (i.e., exactly 90 degree).

   • The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs.

   • Pythagorean Theorem: For a right-angled triangle with legs a and b and hypotenuse c.

3. Obtuse-angled Triangle:

   • Angles: Exactly one interior angle is obtuse (i.e., greater than 90∘ but less than 180∘).

   • The other two angles must be acute.

Formulas and Calculations

• Perimeter: The perimeter (P) of any triangle is the sum of the lengths of its three sides.

  • P=a+b+c

• Area: The area (A) of a triangle can be calculated in several ways:

  • Using Base and Height: If the base (b) and corresponding height (h) are known:

    • A= 0.5*base*height

  • Using Heron's Formula (when all three sides are known):

    • First, calculate the semi-perimeter (s): s=0.5*(a+b+c)

    • Then, A=sqrts(s−a)(s−b)(s−c)

  • Using Two Sides and the Included Angle (Trigonometric Formula): If two sides (a, b) and the angle (∠C) between them are known:

    • A=frac12absin(angleC) (and similar formulas for other side-angle combinations)

Understanding these properties and classifications is fundamental for solving geometric problems and for applications in various scientific and engineering disciplines.